MA 4451: Boundary Value Problems
B-Term, Fall 2026
Department of Mathematical Sciences
Worcester Polytechnic Institute
B-Term, Fall 2026
Department of Mathematical Sciences
Worcester Polytechnic Institute
Instructor: Prof. B.S. Tilley, Department of Mathematical Sciences,
Worcester Polytechnic Institute
Office: SH 101 Phone: (508) 831-6664
e-mail: tilley@wpi.edu
Not read after 7:00 pm;
M-F: reply within 24 hrs; Sat/Sun: 48 hrs
Office Hours (no appointment needed):
T: 2:30 - 3:50 pm, R: 2:00 - 3:30 pm
These times don't work, make an appointment! e-mail tilley@wpi.edu
PLA: Christopher Pacheco
e-mail:
Office Hours: TBA
Lecture Times: MR - 10:00 - 11:50 AM, Room SH 207
Discussion: W - 1:00 - 1:50 PM, Room SL 402
Textbook: Applied Partial Differential Equations, with Fourier Series and Boundary Value Problems, by R. Haberman and P. Choboter, (Pearson) ISBN-13: 9780138240738 (2025 update)
Note: This textbook has 6 editions. Used editions are fine (homework problems are not taken from the text)
Additional References (available on reserve in the WPI Library):
Partial Differential Equations for Scientists and Engineers, by S.J. Farlow (Dover) ISBN: 0-486-67620-X
Boundary Value Problems and Partial Differential Equations, by D.L. Powers (Academic Press) ISBN-13: 978-0123747198
Spectral Methods in Matlab, by L.N. Trefethen (SIAM) ISBN: 0-89871-456-6
Course Description: Science and engineering majors often encounter partial differential equations in the study of heat flow, vibrations, electric circuits and similar areas. Solution techniques for these types of problems will be emphasized in this course. Topics covered include: derivation of partial differential equations as models of prototype problems in the areas mentioned above, Fourier Series, solution of linear partial differential equations by separation of variables, Fourier integrals and a study of Bessel functions.
Provisional Outline of Topics (updated weekly)
Week 1: Review of solutions to linear ODE's, Introduction to Boundary Value Problems, Derivation of Heat Equation, Steady-State Solutions
Week 2: Separation of Variables, Boundary Conditions, Introduction to Fourier Series
Week 3: Properties of Fourier Series, Sturm-Liouville Theory
Week 4: Derivation of the Wave Equation, Boundary Conditions, Nondimensionalization
Week 5: Laplace's Equation, Steady-State Solutions of Heat/Wave Equations in 2D
Week 6: Heat/Wave Equations on Circular Domains, Eigenfunction Expansion, Fourier Transform
Week 7: Review and Final Exam
Class Expectations: As an upper-level mathematics course, collaborative learning and active engagement are expected. Collaborative learning means that students collaborate together to learn the material in the course. Active engagement by students means that students accept the responsibility for their own learning of the material and do not perceive the instructor (professor) as a source of all knowledge. In order to meet these expectations, the classroom environment must be professional and supportive. Students are expected to treat each other with mutual respect, provide constructive feedback to other students, and to realize that as humans we all need guidance at times.
Attendance: While attendance is not taken in lecture and does not count for the grade, students are responsible for all material covered in lecture and need to meet deadlines for homework assignments. Attendance will be taken in the Discussion sections, and for unapproved absences, a five (5) point deduction of that week's homework score will be assessed. Students need prior approval from Prof. Tilley at least 48 hours before the start of the Discussion section for an absence to be excused. Flexible arrangements will be considered on a case-by-case basis.
Use of Generative AI: This course will be Prof. Tilley's first attempt to show how to use these tools to build on the topics that we cover in class, not as a substitute for learning, Some lectures and discussion activities will directly pose queries to Gemini or Copilot, and responses will need to be critically evaluated by the students for mathematical correctness based on the topics of this course. Some of the exercises will leverage these AI's to build tools to visualize the solutions of the problems under consideration. Homework problems will be clearly marked as to whether Generative AI is to be used, and the use of any Generative AI to solve problems that are not labelled in this way is a violation of academic integrity.
Changes in Syllabus: This syllabus is a guide for the course and is subject to change. Any changes to the current syllabus will be sent in writing with at least 1 week notice.
Grading: There are three components that make up the final grade.
Homework: (20% of final grade) Each week, a set of homework problems will be assigned, which are due on the Friday of that week, excluding Thanksgiving break. Included in these problems are the activities which take place during the Wednesday Discussion period. The format of the homework assignment is described here. There are six homework sets, and the top five homework scores are counted in the final grade calculation. Late homework assignments will be graded for up to 50% of the possible points as long as they are submitted before the solutions are posted. Deductions due to a discussion absence will be assessed after the grade of the homework assignment is determined.
Quizzes: (50 % of final grade) Five written quizzes are schedule for the first five Monday's of the term. All quizzes are 15 minutes, are closed-book, no notes, no internet, and are based on the previous week's homework assignment. These questions are intended to see if the student has mastered the fundamental aspects of the assigned topic. All quizzes are counted toward the final grade. If a student cannot attend a quiz, there will be a single make-up quiz on Reading Day, December 7, 2026. The topic of the make-up quiz will be any topic covered in the first six homework assignments. Only students who have missed a quiz or have scored below a 70% on one of the quizzes will be allowed to take the make-up quiz. The grade on the make-up quiz replaces the lowest quiz grade below 70%.
Final Exam : (30% of final grade) December 10, 2026 . This 100-minute written comprehensive exam is open book, open notes, no internet.
Student Accommodations
Students with approved academic accommodations should plan to submit their accommodation letters through the Office of Accessibility Services Student Portal. Should you have any questions about how accommodations can be implemented in this particular course, please contact me as soon as possible. Students who are not currently registered with the Office of Accessibility Services (OAS) but who would like to find out more information about requesting accommodations, documentation guidelines, and what the accommodated interactive process entails should plan to contact OAS either by email: AccessibilityServices@wpi.edu, by phone (508) 831-4908, or by stopping by the office on the 5th floor of Unity Hall.